Partial-dual Euler-genus polynomials for two classes of bouquets

Abstract

[European J. Combin., 2020, 86: Paper No. 103084, 20 pp.] introduced the concept of partial-dual Euler-genus polynomial in the ribbon graphs and gave the interpolation conjecture. That is, the partial-dual Euler-genus polynomial for any non-orientable ribbon graph is interpolating. In fact, [European J. Combin., 2022, 102: Paper No. 103493, 7 pp.] gave two classes of counterexamples to deny the conjecture, and only one or two of the side loops contained in the two classes of bouquets were non-orientable. On the basis of [European J. Combin., 2022, 102: Paper No. 103493, 7 pp.], we further calculate the partial-dual Euler-genus polynomials of two other classes of bouquets. One is non-interpolating, whose side loop has an arbitrary number of non-orientable loops. The other is interpolating, whose side loop has an arbitrary number of both non-orientable loops and orientable loops.

Graphical abstract

Keywords

Ribbon graph / partial-dual / genus / polynomial / interpolating

References

[1] Bollobás B. A polynomial of graphs on surfaces. Math Ann 2002; 323(1): 81–96

[2] Chen Q Y, Chen Y C. Parallel edges in ribbon graphs and interpolating behavior of partial-duality polynomials. European J Combin 2022; 102: 103492

[3] Chmutov S. Generalized duality for graphs on surfaces and the signed Bollobás—Riordan polynomial. J Combin Theory Ser B 2009; 99(3): 617–638

[4] Ellis-Monaghan J A, Moffatt I. Twisted duality for embedded graphs. Trans Amer Math Soc 2012; 364(3): 1529–1569

[5] Ellis-MonaghanJ AMoffattI. Graphs on Surfaces: Dualities, Polynomials, and Knots, SpringerBriefs in Mathematics. New York: Springer, 2013

[6] Gross J L, Mansour T, Tucker T W. Partial duality for ribbon graphs, I: distributions. European J Combin 2020; 86: 103084

[7] Gross J L, Mansour T, Tucker T W. Partial duality for ribbon graphs, II: partial-twuality polynomials and monodromy computations. European J Combin 2021; 95: 103329

[8] LivingstonC. Knot Theory, Carus Mathematical Monographs. Vol 24, Washington, DC: Mathematical Association of America, 1993

[9] Moffatt I. A characterization of partially dual graphs. J Graph Theory 2011; 67(3): 198–217

[10] MunkresJ R. Topología, 2nd Ed, Upper Saddle River. NJ: Prentice Hall, 2000 (in Spanish)

[11] Yan Q, Jin X A. Counterexamples to the interpolating conjecture on partial-dual Genus polynomials of ribbon graphs. European J Combin 2022; 102: 103493

[12] Yan Q, Jin X A. Partial-dual polynomials and signed intersection graphs. Forum Math Sigma 2022; 10: e69

RIGHTS & PERMISSIONS

2024 Higher Education Press 2024